# KIAS Workshop on Combinatorics

The "**KIAS Workshop on Combinatorics**" will be held at KIAS, Seoul on May 30- June 1, 2013. If you want to participate, please write a registration form and email to kias@combinatorics.kr until May 10. This workshop will be supported by Open KIAS.

### Information

**Title**KIAS Workshop on Combinatorics

**Date**May 30 - June 1 (Thu-Sat), 2013**Venue**Room 1503, KIAS

**Organizers**Jeong Han Kim, KIAS

Boram Park, NIMS

**Program Committee**Seog-Jin Kim, Konkuk University

Youngsoo Kwon, Yeungnam University

Seunghyun Seo, Kangwon National University

Heesung Shin, Inha University

**Invited Speakers**Gi-Sang Cheon, Sungkyunkwan University

Jeong Ok Choi. GIST

Mitsugu Hirasaka, Pusan National University

Hyun Kwang Kim, POSTECH

Sangwook Kim, Chonnam National University

Seog-Jin Kim, Konkuk University

Younjin Kim, KAIST

Youngsoo Kwon, Yeungnam University

Seunghyun Seo, Kangwon National University

Hwanchul Yoo, KIAS

We are going to

give 10 invited talks without contributed talks.

provide for 5 meals (dinner of May 30, breakfast, lunch, & dinner of May 31, breakfast of June 1) of all participates.

support the accommodation for two nights of all students who register until April 30.

### Schedule

May 30 (Thursday)

13h00 ~ 14h00 Registration

14h00 ~ 14h10 Opening Ceremony

14h10 ~ 17h20 Session A

14h10 ~ 15h00 Talk 1

15h10 ~ 16h00 Talk 2

16h00 ~ 16h30 Break

16h30 ~ 17h20 Talk 3

May 31 (Friday)

09h30 ~ 11h50 Session B

09h30 ~ 10h20 Talk 4

10h20 ~ 10h50 Break

11h00 ~ 11h50 Talk 5

11h50 ~ 14h00 Lunch

14h10 ~ 17h20 Session C

14h10 ~ 15h00 Talk 6

15h10 ~ 16h00 Talk 7

16h00 ~ 16h30 Break

16h30 ~ 17h20 Talk 8

17h40 ~ Banquet

June 1 (Saturday)

09h30 ~ 11h50 Session D

09h30 ~ 10h20 Talk 9

10h20 ~ 10h50 Break

11h00 ~ 11h50 Talk 10

11h50 ~ Closing Ceremony

Session A

**Chairman**Seog-Jin Kim, Konkuk UniversityTalk 1

**Speaker**Gi-Sang Cheon, Sungkyunkwan University**Title**The Riordan group and related topics in Combinatorics and Matrix Theory**Abstract**The Riordan group is the set of inﬁnite lower triangular matrices whose kth column has the generating function g(z)f(z)^{k}where g and f are elements of the ring of formal power series C[[z]] such that g(0)=1, f(0)=0 and f'(0)<>0. Such a matrix is called Riordan matrix and denoted as (g(z); f(z)) or (g; f). The Riordan group shows up naturally in a variety of combinatorial settings and combinatorial matrix theory. This talk is given by two parts. The concept of Riordan group and Riordan matrix will be introduced in the ﬁrst part by presenting fundamental properties and interesting subgroups. In the second part, we discuss how this concept can be applied to several problems arising in combinatorics and matrix theory.

Talk 2

**Speaker**Seog-Jin Kim, Konkuk University**Title**Coloring of the square of Kneser graphs**Abstract**The Kneser graph K(n,k) is the graph whose vertices are the k-elements subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The square G^{2}of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in G^{2}if the distance between u and v in G is at most 2. We denote the square of the Kneser graph K(n,k) by K^{2}(n,k). The problem of computing χ(K^{2}(n,k)), which was originally posed by Füredi, was introduced and discussed by Kim and Nakprasit (2004). Note that that A and B are adjacent in K^{2}(n,k) if and only if A∩B=∅ or |A∩B|≧3k-n. Therefore, K^{2}(n,k) is the complete graph K_{t}where t=_{n}C_{k}if n≧3k-1, and K^{2}(n,k) is a perfect matching if n=2k. But for 2k+1≦n≦3k-2, the exact value of χ(K^{2}(n,k)) is not known. Hence it is an interesting problem to determine the chromatic number of the square of the Kneser graph K(2k+1,k) as the first nontrivial case. We will give a brief introduction of the problem, and present recent results. This talk is based on joint work with Boram Park.

Talk 3

**Speaker**Hyun Kwang Kim, POSTECH**Title**Polytope numbers and their applications**Abstract**These is an introductory lecture on combinatorial properties of polytope numbers. We ﬁrst introduce polytope numbers and their basic properties such as product formula and decomposition theorems. Next we illustrate these properties with some well-known polytopes. Finally we give research problems related to polytope numbers.

Session B

**Chairman**Youngsoo Kwon, Yeungnam UniversityTalk 4

**Speaker**Jeong Ok Choi, GIST**Title**Fractional weak discrepancy of posets**Abstract**In this talk, various discrepancies of posets (Partially Ordered Sets) will be introduced. In particular, fractional weak discrepancy of a poset is emphasized as a reﬁnement of measuring weakness of a poset. We characterize forbidden structure for posets preventing fractional weak discrepancy larger than k for each natural number k. Also, we give the range of fractional weak discrepancy of (M;2)-free posets.

Talk 5

Speaker Mitsugu Hirasaka, Pusan National University

Title Zeta functions of adjacency algebras of association schemes

Abstrac For a module L which has only finitely many submodules with a given finite index we define the zeta function of L to be a formal Dirichlet series ζ

_{L}(s)=∑_{n≥1 }a_{n }n^{-s}where a_{n}is the number of submodules of L with index n. For a commutative ring R and an association scheme (X,S) we denote the adjacency algebra of (X,S) over R by RS. In this talk we aim to compute ζ_{ZS}(s) under several assumptions where ZS is viewed as a regular ZS-module.

Session C

Chairman Heesung Shin, Inha University

Talk 6

**Speaker**Youngsoo Kwon, Yeungnam University**Title**Classiﬁcation of regular embeddings of some graph families**Abstract**A regular embedding is a highly symmetric graph embedding onto a surface. Classiﬁcations of regular embeddings are pursued by three different directions: by given graphs, groups and surfaces. In this talk, we will consider classiﬁcation of regular embeddings of graphs. I will introduce several methods to classify regular embeddings of graphs and some recent results.

Talk 7

**Speaker**Seunghyun Seo, Kangwon National University**Title**Colored permutations and generalizations of derangements**Abstract**A derangement is a permutation without any ﬁxed points. There are several generalizations of derangements in the literature. In this talk, we introduce 5 types of colored permutations which are all generalizations of derangements. We present the generating function of each type and ﬁnd the hierarchy of them combinatorially. We also present bijective proofs among the objects. This is joint work with Dongsu Kim(NIMS) and Jang Soo Kim(KIAS).

Talk 8

**Speaker**Hwanchul Yoo, KIAS**Title**Balanced labellings of afﬁne permutations and symmetric functions**Abstract**In this talk, we introduce a generalization of the balanced labellings of Fomin, Greene, Reiner, and Shimozono. We extend the notion in two directions: (1) we deﬁne the diagrams of afﬁne permutations and the balanced labellings on them; (2) we deﬁne the set-valued version of the balanced labellings. We show that the column-strict balanced labellings on the diagram of an afﬁne permutation yield the afﬁne Stanley symmetric function deﬁned by Lam, and that the column-strict set-valued balanced labellings yield the afﬁne stable Grothendieck polynomial of Lam. Moreover, once we impose suitable ﬂag conditions, the ﬂagged column-strict set-valued balanced labellings on the diagram of a ﬁnite permutation give a monomial expansion of the Grothendieck polynomial of Lascoux and Schutzenberger. We also give a necessary and sufﬁcient condition for a diagram to be an afﬁne permutation diagram.

Session D

**Chairman**Seunghyun Seo, Kangwon National UniversityTalk 9

**Speaker**Sangwook Kim, Chonnam National University**Title**Flag enumeration of matroid base polytopes**Abstract**For a matroid on [n], a matroid base polytope is the polytope in R^{n}whose vertices are the incidence vectors of the bases of the matroid. In this talk, we discuss ﬂag information of matroid base polytopes for some classes of matroids such as rank 2 matroids and lattice path matroids

Talk 10

**Speaker**Younjin Kim, KAIST**Title**On Combinatorial problems of Erdös**Abstract**For a property Γ and a family of sets F, let f(F,Γ) be the size of the largest subfamily of F having property Γ. For a positive integer m, let f(m,Γ) be the minimum of f(F,Γ) over all families of size m. In 1972, Erdös and Shelah also considered Γ to be the property that no four distinct sets satisfy F_{1}∩F_{2}=F_{3}and F_{1}∩F_{2}=F_{4}. Such families are called B_{2}-free. Erdös and Shelah gave an example showing f(m,B_{2}-free)≦(3/2)m^{2/3}and they also conjectured f(m,B_{2}-free)>c_{2}m^{2/3}. We verify a conjecture of Erdös and Shelah from 1972. In1964, Erdös, Hajnal, and Moon introduced the following problem: get the minimum size of a graph G such that G does not contain F as a subgraph but the addition of any new edge creates at least one copy of F in G. This minimum is called the saturation number of F. We obtain the saturation number of C_{k}, where C_{k}is a cycle with length k.

### Travel Information

Please go to webpage: http://kor.kias.re.kr/sub06/sub06_03.jsp for detail information.

### Banquet (Dinner of May 31)

장소는 홀리데이인 성북 호텔 (http://www.holiday.co.kr/holiday/) 뷔페 입니다.

호텔측에서 준비한 차량으로 이동할 예정입니다. (5:30분 고등과학원앞)

### Accommodation

대부분의 참가자들에게 키아스내의 숙소(Kiastel 과 기숙사)를 제공할 예정입니다.

키아스내의 충분한 숙소확보가 어려운 관계로 대학원생중의 일부는 키아스 외부 숙소로 배정되었습니다. 예정은 홀리데이인 성북 호텔에 2인 1실로 제공 예정이었으나 참가자들의 편의를 위해 도보로 이동할 수 있는 거리에 있는 곳으로 변경하여 1인 1실로 숙소로 배정하였습니다. 숙소는 co-up(http://rent.co-op.co.kr/accommodations_16.htm) 입니다. 해당대학원생분들께 안내메일을 드렸습니다.

### Registration

If you want to participate, please fill out the registration form at the bottom of this page and email to kias@combinatorics.kr until May 10, 2013.

We plan to reserve hotels near KIAS for participants who register before April 30.

We are going to support the accommodation of graduate students who registered on before April 30.

We encourage students to come to the workshop.

### Registration Form (for KIAS Workshp on Combinatorics)

English Name:

Original Name (usually Korean):

Affiliation:

Status: (Professor/ Post-Doc/ Researcher/ Student)

Gender: (Male/Female)

If you are a (graduate) student, do you want a support accommodation? (Y/N)

Accommodation on May 30: (Y/N)

Accommodation on May 31: (Y/N)

Notice that we support the accommodation for two nights of any student who registered BEFORE APRIL 30.

If you are not a graduate student, do you want a reservation of a hotel? (Y/N)

Accommodation on May 30: (Y/N)

Accommodation on May 31: (Y/N)

Notice that we reserve the accommodation of participant who registered BEFORE APRIL 30.